Homomorphisms of the Lie algebras associated with a symplectic manifold

C ^OMPOSITIO M ^ATHEMATICA

C. J. A ^TKIN J. G ^RABOWSKI

Homomorphisms of the Lie algebras associated with a symplectic manifold

Compositio Mathematica, tome 76, n^o3 (1990), p. 315-349

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Homomorphisms

^{of the Lie}

algebras

associated with a

symplectic

manifold

C. J. ATKIN and J. GRABOWSKI

1Department of Mathematics, ^Victoria University of Wellington, ^{P.O. Box 600,} Wellington, ^New Zealand, 2J. Grabowski, Instytut Matematyki U.W., P.K. i N. IXp. 00-901 Warszawa, Poland

© 1990 Kluwer Academic Publishers. Printed in the Netherlands.

Received 5 November 1988; accepted ^{in revised form 30} October 1989

1. Introduction

We have each given (in [1] ^and [7]) proofs ^{of an} algebraic result - restated for

our present purposes ^as (6.2) below - which in effect (see (6.5)) constructs a one- one correspondence between the points ^{of a} symplectic manifold and certain

subalgebras of its Lie algebra of Poisson brackets. Our aim here is, firstly, ^to

extend this result to the Lie algebras ^of locally, globally, ^and conformally

Hamiltonian vector fields determined by ^the symplectic structure; and then to utilise it to prove that each of these algebras determines the manifold, ^{as far as}

that is possible. ^In fact, ^we approach ^these ’uniqueness ^theorems’ by studying

certain types ^{of Lie} homomorphism (which ^we classify in Section 7) ^between

such Lie algebras, rather than by reconstructing the manifold from the algebra;

this method (modelled ^{on that in} [6]) ^both gives ^and requires less structural

information, ^but yields ^more facts about homomorphisms. Indeed, ^{we have}

taken no pains ^{to delve more} deeply ^{into the structure of our} algebras ^{than our} techniques demand, ^{and those} techniques ^are perhaps ^more interesting ^{than the}

results which motivated them.

In Section 2 ^we present ^some definitions and facts not related to symplectic

structures; Section 3 introduces the notion of n-ample algebras ^{of vector fields.}

In Sections 4 and 5 we review some definitions and notations, ^and give proofs ^of

some properties which will be needed subsequently (and ^{one or two which will}

not, such ^as (5.8)). ^{Here we} mostly follow, and often refer to, the well-known paper [2]. Although ^{we allow both the} real-analytic ^{and the} holomorphic (Stein) differentiability classes, the results from [2] ^{which we} quote ^are merely local,

and as such hold in these cases without _any modification. Then Section 6 gives

the algebraic characterisations of the points ^{of a} symplectic manifold, ^{Section 7}

classifies suitable homomorphisms, and Section 8 considers the application ^to epimorphisms ^and isomorphisms.

Special ^{cases of the} ’uniqueness theorem’ have been proved ^before (though

not, ^we believe, published). ^Our method, however, ^{seems to be} the first which

applies simultaneously ^{to so} many ^cases.

2. Preliminaries on manifolds and on Lie algebras

(2.1 ) ^In speaking of an n-dimensional manifold M of dass W over the field F, ^we shall mean one of three things:

(a) ^{a real} paracompact manifold of differentiability ^class Coo and of real

dimension n, when W denotes Coo and F denotes the real field R;

(b) ^{a real} paracompact ^{manifold of} differentiability class C03C9 and of real dimension n (when W denotes C03C9 and F denotes R);

(c) ^a complex manifold of complex dimension n for which each connected component ^{is Stein} (when W denotes the holomorphic differentiability ^class

Jf and F denotes the complex ^field C).

We shall not consider complex ^{manifolds whose} components ^{are not} Stein,

and shall often omit explicit ^{mention of} Y, F, ^{or n.}

(2.2) ^{For each of} (a), (b), (c), ^{one has an} embedding ^theorem (due ^to Whitney [ 14] ^{in case} (a), ^{to Remmert and to} Narasimhan [12] ^{in case} (c), ^{and to Grauert} [8] ^{in case} (b)): ^a connected manifold of dass W and dimension n is CC-

diffeomorphic ^{to a} closed Y-submanifold of Fln + 1. (Note ^that by ^{a ’closed}

submanifold’ we understand ^a ’properly ^and regularly ^embedded submanifold’.) (2.3) ^{For a manifold M of class} W, ^{we denote} by ^{TM the bundle of} tangent

vectors (meaning, ^{in case} (c), ^the tangent ^vectors of type (1, )), ^and by ^{T*M the} corresponding cotangent bundle. The vector space ^over F of exterior forms of dass W on M (again, ^{in case} (c), these forms are to be of type (k, 0) ^and holomorphic) will be called

S2k(M).

The exterior derivative

d:03A9k(M) ~ 03A9k+1(M)

is defined as usual; ^we write its kernel as

Z’(M)

^{and its} image ^as Bk+ l(M), ^with

the convention that

03A9k(M)

^{= 0 when k 0. The chain} complex

(03A9k(M),

d) ^{is the}

de Rham complex ^{of M.}

(2.4) ^{LEMMA. Let M be a} manifold of class W; ^let p, q ^E M, p ~ q, and let k be a

nonnegative integer. ^Given any k-jets (of ^F-valued functions) ^at p and _q, there

exists f

en’(M)

which has those k-jets.

Proof. ^{When M =} F", this is trivial. (2.2) ^then gives ^{it in} general.

(2.5) ^For each of the cases of (2.1), ^{there is a de Rham} isomorphism

where the cohomology may conveniently be assumed singular. ^{For case} (a), ^this

is de Rham’s theorem. For case (c), ^{it is a} well-known consequence of Cartan’s Theorem B; ^{see, for} instance, [4], exposé XX, ^or p. 80 of [9]. ^{The same} argument

may be applied ^{in case} (b), ^where Tognoli [13] ^has pointed ^{out the} validity ^of

Theorems A and B in a real-analytic ^version.

(2.6) PROPOSITION. Let M be a manifold of class CC and dimension n; take

m ⁼ 2n + 1. Then there exist F-valued functions x1,..., Xm of ^{class CC on M, such} that, for any integer ^{k ~ 0 and} any form 03C8 ^E

0’(M),

^{there exist} F-valued functions fi1,i2,....ik of class W ^{on M} ( for ^all i1, i2, ... , ik ^with 1il i2 ^... ik ^~ m), for

which

Proof. ^{It will} clearly ^{suffice to} prove the result for each component ^{of M} individually. ^{So we may suppose} (see (2.2)) ^{that M} is Y-embedded in Fm, ^with embedding j : ^{M ~ Fm.} The natural monomorphism TM ~ j*TFm ^defined by Tj

dualises to epimorphisms 03C0~: A

~T*Fm|j(M) ~

^{A tT* M for} any ?. Let

!)k(M),

!)k(Fm)

denote the sheaves of germs of k-forms of class ^on M, ^Fm respectively;

then _nk induces a sheaf epimorphism (over M)

Let Q ^{= ker} J, ^so that there is an exact sequence

Let Cm ^{be the structure sheaf over M} of germs of COO functions in ^case (a), ^C03C9

functions in case (b), ^and holomorphic ^{functions in case} (c). ^{Then J is a} homomorphism ^of (9,-modules, ^{so that} Q ^{is also an} (9,-module.

Let Y1, ..., y. be the coordinate functions on Fm. Then

fl’(F’)

^{is free over the} appropriate ^{structure sheaf} (9F-; indeed, it has free generators given by ^the

sections dyi ^{^ ··· A} dytk ^{for 1 ~ i ... ik ~ m. Hence,} as j*OFm ⁼ (9, ^trivi- ally,

j*03A9k(Fm)

^{is free over} (9m, and it has free generators (dyi ^{^ ··· A} dyi,)-

induced from the sections

In the C03C9 and holomorphic ^cases

!)k(M)

^{is coherent over} OM, ^{and so of course}

is the free OM-module

j*03A9k(Fm).

By ^Serre’s 3-lemma, then, Q is also coherent

over (9m. From Theorem B (see (2.5)) ^{we know that}

Hk(Q)

^{= 0 for k} &#x3E; 1.

In the COO _case, (9m ^{is soft} (see [3] ^or [5]) ^{so that} Q is also soft and

H1(Q)

^{= 0.}

In all three _cases, the cohomology ^exact sequence of (1)

leads to the result that J* ^{is onto. The} given form 03C8 ^e

03A9k(M)

^is consequently ^the

image J* ^{of a} section 0

ofj*fik(Fm);

but, ^{as remarked at}

(2),j*!)k(Fm)

^{is free over} (9m, ^so that there are

sections fi1...ik

^{of (9, for which}

Applying J* (that ^is, compounding ^with J), ^we obtain after some bookkeeping

where, ^{for each} i, xi = Yi 0 j; ^{this is} clearly the result.

(2.7) Again ^{let M be a manifold} of class Y. Then r(M) ^{will denote the Lie} algebra

of sections of TM of class Y. If 5F is a foliation of M of dass W, ^let 0393(F) ^{be the}

Lie subalgebra ^of r(M) consisting ^{of vector fields} everywhere tangent ^{to 97. In} general, ^{if K is a vector} subspace ^of r(M) ^and p ^E M, ^set

(2.8) ^{If oc E}

S2k(M),

^let ro(a) denote the class of vector fields of class which leave

oc invariant, ^{and let} r(a) be the class of vector fields which operate ^{on a as} multiplication by ^a locally ^constant function. In other words, if £f denotes the Lie derivative,

As Y[X,Y] = [YX, 2 y], ^{it follows that}

so that 0393(03B1) ^{is a Lie} subalgebra of r(M) ^and 03930(03B1) ^a Lie ideal in r(a) (including ^its commutator).

(2.9) ^For any Lie algebra ^{L over} F, ^let Z(L) ^denote the class of all self-

normalising ^maximal proper finite-codimensional Lie subalgebras ^{of L.} (Notice

that a maximal subalgebra ^is self-normalising ^{if and} only ^{if it is not an} ideal, ^and

that it can be an ideal if and only if it includes the commutator.)

We may call 1:(L) ^the ’spectrum’ ^{of L.}

Let L(n) denote the nth. derived ideal of L, ^{for n =} 0, 1, 2, ... ; ^{thus L(0) =} L and,

for each _n, L(n+1) ⁼ [L (n),

L(n)].

It will be convenient for technical reasons to

define the ’n-spectrum’ E"(L) (for n ⁼ 1, 2, 3, ... ) ^as the class of subalgebras Q ^of

L such that Q ^E 03A3(L) ^{and also}

We have already observed that

Since L(n) is an ideal in L, Q ^{+ L(n) is a} subalgebra; ^{thus for} Q E 1:(L), (1) ^is equivalent ^to

(2.10) LEMMA. Let 4): L1 ~ L2 ^{be a} surjective homomorphism of ^Lie algebras

over F. Then, for any positive integer ^n,

Proof. Certainly

03A6(q’), 03A6-1(Q)

^are finite-codimensional Lie subalgebras ^of L2, L 1 respectively. ^{Let R be a Lie} subalgebra ^of L2 ^{such that R ;:2} 03A6(Q’).

Then 03A6-1(R) ~ Q’;

consequently,

either 03A6-1(R) = Q’ or 03A6-1(R) = L1,

and,

as 03A6 is surjective, ^{R =}

03A6(03A6-1(R))

^{is either} O(Q’) ^or O(Li) ⁼ L2. ^Hence O(Q’)

is either L2 ^{or a} maximal proper Lie subalgebra. Similarly, ^{let S be a Lie} subalgebra ^of L1 such that S ~

03A6-1(Q);

^{then as 8;:2}

(D - ’(0), S = 03A6-1(03A6(S)).

But 03A6(S) ~ Q. ^Therefore 0(5) ⁼ Q ^or 0(5) ⁼ L2, ^{and either S}

= 03A6-1(Q)

^or

S ⁼ Li;

hence 03A6-1(Q)

^{is a maximal} proper Lie subalgebra ^of L1.

Now, if

03A6-1(Q)

^{were a} proper Lie ideal of L1, Q ⁼

03A6(03A6-1(Q))

^{would be a}

proper Lie ideal of L2, since 03A6 is surjective; ^if 03A6(Q’) ^{were a} proper Lie ideal of L2,

03A6-1(03A6(Q"))

^{would be a} proper Lie ideal in L1 including Q’, ^{and therefore would}

be equal ^to Q’. ^{But neither} Q ^nor Q’ ^{is a Lie ideal} (in L2, L1 respectively); ^so

03A6-1(Q)

^and O(Q’) ^{are not} Lie ideals. This _proves (a) ^and (b) ^{when n = 1.}

Finally, suppose ⁿ &#x3E; 1. Then, as 03A6 is epimorphic,

whilst, ^if

03A6-1(Q) ~ Lln),

^then Q ⁼

03A6(03A6-1(Q)) ~ 03A6(L(n)1)

⁼

L(n)2.

The results now

follow, by (2.9)(3) ^and (2.9)(1) respectively.

(2.11) ^{LEMMA. Let L be a Lie} algebra ^over F, ^{and let K be a Lie ideal} of ^L including ^L(1).

Then

(a) if (L, K] ^{= K(1) and} Q E 1:(L), ^{there exists} NeE(K) ^{such that} Q ^{n K z} N;

(b) if, for ^some positive integer ^n, Q ^{E yn 1}

l(L),

^{there exists N E} 03A3n(K) ^{such that} Q ^{n K z N.}

Proof. ^{Consider case} (a). ^{As K is an} ideal, ^{K +} Q ^{is a Lie} subalgebra ^{of L.}

However, K e Q; ^for otherwise, ^{as K ~} L(1), ^we should have Q ;:2 ^{L(1) and} Q

would be an ideal. Hence K + Q ~ Q and, by maximality, ^{K +} Q ^{= L.}

By hypothesis, [Q, ^{K] ~} [L, K] ^{= K(’).} Thus, ^{if K(1) ~} Q, ^{K must normalise} Q ^and (as ^{K +} Q ⁼ L) Q ^{must be an ideal in L. This is} false, ^as Q ^E Z(L); ^we

deduce that

In case (b), K(n) ~ L(n+1) and, ^as Q ^E

03A3n+1(L),

^{it follows} immediately ^that

In either case, K(1) + Q ^{is a} subalgebra (as ^{K(1) is an} ideal) ^{which is not} equal

to Q; ^{thus K(’) +} Q ⁼ L, ^{and as a} consequence

As K ~ Q, ^{K n} Q is of finite positive codimension in K. Let N be ^a maximal proper subalgebra ^{of K} including Q ^{n K} (which ^we may construct by ^finite induction). Then, by (2), ^{K(’) + N =} K; in view of the maximality ^of N, ^this implies ^that N ";/2 ^{K(1) and N is not a} Lie ideal in K (see (2.9)). ^{This proves} (a). ^For (b), observe that (1) gives

exactly ^as (0) ^{led to} (2). Ergo, ^{K(") + N =} K, ^{which shows N E} En(K), by (2.9)(3).

3. Lie algebras ^{of vector fields}

(3.1) ^Once more, let M be a manifold of class W and let L be a Lie subalgebra (over F) ^of r(M) (see (2.7)). ^{We shall say that L is} n-ample, where n is ^a positive integer ^{- more} precisely, ^{L is an} n-ample subalgebra ^of r(M) - if, ^for each p ^{E M,} Lp E 03A3n(L) (see (2.9)).

(3.2) ^LEMMA. (a) ^{Let L be an} n-ample subalgebra of r(M). ^Then

(b) Suppose ^{L is} a 1-ample subalgebra of r(M) ^and satisfies (1). ^{Then L is n-} ample.

Proof. (a) By hypothesis, Lp ^{+ L(") = L} (see (2.9)(3)). The result follows, ^as

L,(p)

^{= 0 by} definition.

(b) ^If L(n)(p) ⁼ L(p), ^then evidently Lp ^{+ L(n) = L.} Apply (2.9)(3).

(3.3) ^LEMMA. Suppose L1, L2 ^{are Lie} subalgebras of r(M) ^{such that} L1 £; L2 and, for any p ~ M, L1(p) ⁼ L2(p). Then, if L1 ^is n-ample, ^{so is} L2.

Proof. ^Take p E M, ^{and let R be a Lie} subalgebra of L2 which includes

(L2)p.

Then R n L1 ;:2

(L1)p,

^{and, as L1 is} 1-ample, ^either R n L1 ⁼ L, ^or

R n L1 ⁼

(L1)p.

^{As L1( p) =} L2(p), certainly

Consequently

If R ~ L1 = Li, (2) ^and (1) ^{show that R =} L2 ; whilst, if R ~ L1 =

(L1)p,

^{(2) shows}

that R ⁼

(L2)p.

This establishes the maximality ^{of the} subalgebra

(L2)p

^{of L2,}

and it is evidently of finite codimension therein. If it were an ideal in L2,

(L1)p

^{= Li n}

(L2)p

^{would be an ideal in Ll, which it is not.} This shows that, if L1

is 1-ample, ^{so is} L2. ^If Li ^is n-ample, by (3.2)(a)

so that L2 ^is n-ample by (3.2)(b).

4. Symplectic structures and the associated Lie algebras

(4.1) ^A symplectic ^manifold (M, co) of class W and dimension 2n is a manifold M of class and dimension 2n, furnished with an everywhere non-degenerate

closed 2-form 03C9 E

Z2(M).

Following [2], ^{p. 2, we have then a bundle} isomorph-

ism of class Y,

(where ^{i denotes} the internal product), which induces isomorphisms, ^also

denoted by 03BC03C9, of the tensor bundles and their spaces of sections.

(4.2) ^We have also A ⁼ Aro ⁼

i(03BC-103C9(03C9)):03A9r+2(M) ~

gr(M) (ibid., p. 3).

(4.3) ^Let L*(co) = y,

1(B1(M))

^and L(co) =

03BC-103C9(Z1(M))

^denote respectively ^the

spaces of globally ^{and of} locally Hamiltonian vector fields on M. Certainly

so that L(03C9) ^and L*(cv) ^{are Lie} subalgebras ^{of the} algebra ^{of vector} fields, ^and (recall (2.8)) [0393(03C9), 0393(03C9)] ~ Iro(co) ⁼ L(cv) «3. 1) ^on p. 6 of [2]). Following ([2], p.

11), ^we describe fields in 0393(03C9) ^as ’conformally Hamiltonian’.

(4.4) ^{A foliation F} (of ^class W) ^{of the} symplectic ^manifold (M, w) of dass W will itself be described as ’symplectic’ ^{if w restricts to an} everywhere non-degenerate

form on each leaf of ,9’. Thus the leaves also become symplectic ^manifolds.

Similarly, ^a subbundle S (of ^class Y) ^{of TM is} ’symplectic’ ^{if cv restricts to a non-} degenerate ^{form on each fibre of S.} Clearly ^there is the usual correspondence

between symplectic foliations and integrable symplectic subbundles.

(4.5) Given f, ^{g E}

QO(M),

^{one defines} the Poisson bracket (relative ^to w) by

by ^{an easy} computation. This makes

Q°(M)

^{into a Lie} algebra, ^{which we denote} by A(M). Using ^the non-degeneracy ^{of w and} (2.4), ^{one sees that the centre} C(M)

of A(M) consists of the locally ^constant functions of dass W on M; ^that is,

Thus d induces a linear isomorphism A(M)/C(M) ~

B1(M),

which determines

a Lie algebra ^{structure on}

B1(M);

then 03BC03C9: L*(03C9) ^-

B 1 (M)

^{is a Lie} algebra isomorphism. ^{There is a Lie} algebra ^exact sequence

(4.6) ^For X, Y ~ L(03C9), ^{one has} ([2], (3.3), p. 7)

(4.7) ^{The Lie} algebra A(M) ^{is also a} commutative associative algebra ^under pointwise multiplication ^of functions, ^which is related to the Lie algebra

structure by the structural equation ^{derived from} (4.5)(1)

(4.8) PROPOSITION. _{For every} n ~ ^{1 and} every p E M,

Proof. ^{To avoid messy} calculations, ^{let us use} the theorem of Darboux to construct coordinates (xl, ... x2n) of class W on a neighbourhood of p ^{such that}

03C9 is represented ^{on that} neighbourhood ^as

03A3ni=1 dxi

^{A dXn+i’ In these coor-}

dinates,

y. 1

has the form

thus, ^{in terms of} principal parts,

03BC-103C9

^is represented by ^{a constant linear} isomorphism

For a scalar-valued function f, ^{and x in the chart in} question, df(x) ^is represented by the derivative Df(x) ^E

(F2n)*;

^thus J(Df(x)) represents MI,

1(df)(x),

and the kth.

derivative in these coordinates of XI = 03BC-103C9

1(df)

^at p is obtained by identifying

Dk+1f(p) ~ (~k+1F2n)*

^{with a linear} mapping ~kF2n ~

(F2")*

^and compounding

with J.

(A) ^{We now claim} inductively that, given integers k &#x3E;_ 0,0 ~ ~ ~ k, ^{and nonzero}

vectors 03BE ~ ~~ F2n, ~~F2n,

there exists

XE(L*(w))(n)

^{such that} D"X(p) ^{= 0 when} r ~ k ^and r ~ ~, ^whilst

D~X(p). 03BE

⁼ il. For n ⁼ 0, ^{take X =}

X f,

^{where D1(p) = 0}

for r k + 1 and r ~ ~ ⁺ 1, ^and

D~+1f(p)

^{is a} symmetric ^element

of (~~+1F2n)*

such that De +

1f(p) ·

(03BE ^Q r) ⁼

J-1(~) ·

ⁱ for each element T of ^a basis of F2". The existence of such ^a symmetric multilinear map is ^an elementary ^exercise using multinomials; the existence of a suitable f ^E A(M) follows from (2.4).

Suppose the claim established for arbitrary k, ?, 03BE, ~ ^and given n ~ 0, ^{and take}

X E

(L*(03C9))(n)

^{so that} X(p) :0 0, DX(p) ⁼ 0,

D2X(p)

⁼ 0,..., ^Dk+

1X(p)

^{= 0. Then}

for _any Y E r(M), the local coordinate representations ^of [X, Y] ^are

and so on to

Dk[X, Y](p) = Dk+1Y(p)·(,

X(p)) + terms combining ^lower

derivatives of Y and higher derivatives (to ^{order k +} 1) ^of X; by ^{choice of} X, then,

If Y is chosen in

(L*(w))(n)

^{so that D}

Y(p)

= 0,...,

Dty(p)

⁼ 0,

D~+2Y(p)

⁼ 0, ...

Dk+1 Y(p) ^{= 0, and}

D~+1Y(p) · (03BE

0 X( p)) tl, it follows that [X, Y] satisfies the

required conditions, ^{and of course it is an element}

of (L*(03C9))(n+1).

This proves the claim.

Now take k = e ⁼ 0 and the Proposition ^follows.

Note. The result when M is real follows instantly ^from (5.5). ^{In the} complex

case, however, ^{the above} argument ^{seems the} simplest (though ^{not the most} conclusive) available, ^{and is} largely needed anyway for the ^next result.

(4.9) ^{LEMMA. Let} X E r(M), p E M, ^and X(p) ~ 0; suppose B is ^a subspace of r(M) which includes L*(w). ^{Then the} subspace of ^B

is of infinite codimension in B.

Proof. Use the coordinate system introduced in the proof ^of (4.8). Applying (4.8)(1) inductively, ^we find that in terms of these coordinates

+ terms in lower-order derivatives of Y at p;

but, by the assertion (4.8)(A), there exists for each n ~ ^{1 a field} Y E L*(cv) ^such

that Y( p) = 0, D Y( p) = 0,..., ^D" Y( p) = 0, ^{Dn+ 1} Y(p) · (X(p), X(p), ^..., X(p)) ~ ^0.

Thus

YjXY(p)

^{= 0 for} 0 ~ j n, whilst YnXY(p) ~ ^{0. This} evidently proves the result.

5. Commutators in the Lie algebras

(5.1) ^{In this} section, (M, cv) ^{is a fixed} symplectic ^{manifold of class and}

dimension 2n, ^{and we} abbreviate the previous ^notations (from §§2, 4) A(M), Qr(M), zr(M), Br(M), J.lro’ Aro, [ , ]03C9, to A, 03A9r, Zr, Br, 03BC, , [ , ] respectively. As

in [2], p. 2, set ~ ⁼ wn/n! ^E Q2n, ^the symplectic volume form. We have

This follows

instantly

^from L(cv) ⁼ 03930(03C9) (see (4.3)).

The symplectic adjunction operator ^{*, defined on} p. 2 of [2], ^satisfies

For the symplectic coderivative £5: 03A9p ~ 03C9P-1: a ^~ ( -1)p ^{* d * a, one has}

as is proved ⁱⁿ [10] ^and quoted ^as (1.8)(a) ^on p. 3 of [2].

(5.2) ^THEOREM.

Moreover, ^{there exist elements} xl, X2, ^..., Xm of ^A (where ^{k = 4n +} 1) ^{such that}

the mapping

is surjective.

Proof. Suppose first that _g, h E A. Then, by (4.5)(2) ^and (5.1)(1),

(i denotes the interior product, ^{and we} have used the fact that Q2n ⁼

Z2n).

^Hence

[g, h]~ ~ B2n.

For the converse, suppose f ~ A

and f~ ~ B2n.

^So there exists

03B2~03A92n-1

^such that fri ⁼ d03B2, ^and consequently

Let _{x1, x2,} ... , xm be as in (2.6) (with the difference that the dimension of M is

now 2n). Hence, ^as

*/3 E QI,

there exist functions fI’ f2, ^..., fm ^{E A such that}

It follows that

by (5.1)(3), since A has degree - 2

by (4.5)(1). This shows that f ^E A(1), ^and completes ^the proof.

(5.3) COROLLARY. There is a canonical isomorphism A/A(1) ~

H 2n (M;

F).

Proof. The map ^{A --+} S22": f - fq ^{is an} isomorphism; by (5.2), ^{it carries A(1) on}

to B 2n . As 03A92n = z2n, ^the result follows.

(5.4) NOTES. Both A and

H2n(M;

F) ^{are direct} products ^{of the} corresponding

functors of the individual components ^{of M. When M is real} (whether ^{COO or} cro),

1 will define the fundamental class of each compact component; ^{for each} noncompact component, ^the top cohomology vanishes. Thus the isomorphism

A ~ 03A92n:f ~ f~

^of (5.3) induces, via inclusion C0 ~ A ^and projection 03A92n ~

H2"(M;

F), ^an isomorphism ^{of the} space Co ^of locally ^{constant F-valued}

functions (COO ^{or C03C9 as} appropriate) ^of compact support ^{on M with}

H2n(M;

F).

Since Co ^is clearly ^an abelian ideal of A, (5.3) ^now yields ^{a Lie direct sum} decomposition ^{A =} Co (3 A(1). As Co ^is central, ^{one deduces in turn that} A(1) ⁼ A(2).

These arguments ^{do not} hold in the complex ^case (2.1)(c). ^{In that case, the} top dimension for F-cohomology, namely 2n, ^is only the middle topological dimension, and, ^even for connected M, ^the dimension of H 2n(M; F) may be any finite integer ^or countably ^infinite. Indeed, ^{let E} be any discrete subset of C, ^and

endow M ⁼ (CBE) ^X

(CB{0})2n-1

with the trivial symplectic ^structure (as ^a

subset of

C2n).

The 2n th Betti number of M is the number of points ^of E, ^{and M}

is clearly ^Stein.

Surprisingly ^few examples ^of compact ^real symplectic ^{manifolds are known.}

For a fairly recent, though inconclusive, survey, ^see [11]. There has been some

progress since (by Gromov and McDuff in particular).

(5.5) ^{LEMMA. Let} M be a real symplectic manifold, of ^{class COO or} C03C9. Then

Proof. By (4.5)(4), L*(03C9) ~ A/C(M); by (5.4), ^{A =}

C0 ~ A(1),

^where

Co - C(M). ^Hence [L*(cv), L*(cv)] ⁼ L*(03C9). This suffices (see (4.3)).

(5.6) ^{LEMMA. Let M be a real} symplectic manifold of class COO or C03C9, ^and

suppose the symplectic form ^{w is exact. Then}

Furthermore, 0393(03C9) ^is the semi-direct product of ^its derived ideal L(co) ^{with an}

abelian subalgebra isomorphic ^to C(M).

Proof. ^See [2], p. 12. (We ^{write r} instead of L5, ^{w in} place ^of F; (5.5) ^{must be}

invoked in the CW case, and our formulation allows ^{M not to} be connected).

(5.7) ^{LEMMA. Let M be a} symplectic manifold of class W. Then

(Compare (5.5)).

Proof. ^{In view} of (4.5)(4), [L*(a», L*(cv)] consists of the images ^under

03BC-1d

^of

elements of A(1), ^{which are} characterized by (5.2). Suppose XE L(w) ^and

Y E L*(cv), ^{so that Y =}

03BC-1(df)

^{for some} f ^{E A and} !£ xw ⁼ !£ yw ^{= 0} (see (4.3)).

Hence, by (4.6),

and it is enough, by (5.2), ^{to show that} A(p(X) ^A df)~ ^{E B2n.}

Now

But

This completes ^the proof.

(5.8) ^{LEMMA. Let M} be a connected symplectic manifold of class CC, ^and suppose the symplectic form ^{co is not exact. Then} 0393(03C9) ⁼ L(cv) ⁼ 03930(03C9).

Proof. ^See p. 11 of [2] (after (5.4)); ^no change is needed.

Note that when M is compact, ^{cv cannot be exact} (as ri ^is not).

6. Spectra

(6.1) By ^{a Poisson} algebra ^{over the field} F, ^{we mean a} commutative associative

algebra ^A furnished with ^an F-bilinear operation

with respect ^{to which} it is also a Lie algebra ^over F, and satisfies the structural relation

for all f, g, ^{h ~ A.} Thus, ⁱⁿ particular, A(M) ^{is a Poisson} algebra ^{over F when} (M, cv) ^{is a} symplectic manifold of class 16 (see (4.7)).

Given a Poisson algebra A ^{over F, let} m(A) ^{denote the set of} all maximal finite-codimensional proper associative ideals J in A, ^{and let} E(A) ^{be as in} (2.9).

The theorem which follows is proved (in superficially ^different formulations) ⁱⁿ [7] ^{and in} [1].

(6.2) ^THEOREM. Suppose the Poisson algebra ^A satisfies:

(a) ^{A2 =} A,

(b) for any J ~ m(A), ^{the Lie normaliser}

is a proper finite-codimensional ^linear subspace of ^A.

Then the mapping ^{J ~} 91 A(J) establishes a one-one correspondence ^between m(A) ^and E(A).

(6.3) ^{Now let} (M, 03C9) ^{be a} symplectic manifold of class CC and positive ^dimension.

Given _p ^E M, define

On _p. 17 of [6] ^{it is} proved that the map p H p* ^{furnishes a} bijection ^{between M}

and m(A(M)). ^In addition, (6.4) ^LEMMA.

Proof. Suppose g E A(M) ^and dg(p) = 0; then, by (4.5)(1), [f, g]03C9 ~ p*. ^This

proves that N(M, p) ^~

91A(M)(P*)’

Conversely, suppose

g ~ nA(M)(p*)

^but dg( p) ^{= 0. As 03C9 is} non-degenerate at p, there exists _XE

T*pM

^{for which}

By (2.4), ^there exists f ^E p* ^with df(p) ⁼ x. Thus [ f, g]03C9(P) ~ ⁰ and g e

nA(M)(P*).

This completes ^the proof.

(6.5) ^THEOREM. The _map p H N(M, p) constitutes a bijection of ^{M with} 1:(A(M)).

Proof. ^As remarked in (6.3), p ~ p*: ^{M -} 03A3(A(M)) ^is bijective; hypothesis (6.2)(a) ^is trivial, ^and (6.2)(b) follows from (6.4). Thus the result follows from (6.2)

and (6.4).